Example 69.2 Basketball Data
The data in this example are rankings of 35 college basketball teams. The rankings were made before the start of the 1985–86 season by 10 news services.
The purpose of the principal component analysis is to compute a single variable that best summarizes all 10 of the preseason rankings.
Note that the various news services rank different numbers of teams, varying from 20 through 30 (there is a missing rank in one of the variables, WashPost). And, of course, not all services rank the same teams, so there are missing values in these data. Each of the 35 teams is ranked by at least one news service.
The PRINCOMP procedure omits observations with missing values. To obtain principal component scores for all of the teams, it is necessary to replace the missing values. Since it is the best teams that are ranked, it is not appropriate to replace missing values with the mean of the nonmissing values. Instead, an ad hoc method is used that replaces missing values with the mean of the unassigned ranks. For example, if 20 teams are ranked by a news service, then ranks 21 through 35 are unassigned. The mean of ranks 21 through 35 is 28, so missing values for that variable are replaced by the value 28. To prevent the method of missing-value replacement from having an undue effect on the analysis, each observation is weighted according to the number of nonmissing values it has. See Example 70.2 in
Chapter 70,
The PRINQUAL Procedure,
for an alternative analysis of these data.
Since the first principal component accounts for 78% of the variance, there is substantial agreement among the rankings. The eigenvector shows that all the news services are about equally weighted; this is also suggested by the nearly horizontal line of the pattern profile plot in Output 69.2.3. So a simple average would work almost as well as the first principal component. The following statements produce Output 69.2.1.
/*-----------------------------------------------------------*/
/* */
/* Pre-season 1985 College Basketball Rankings */
/* (rankings of 35 teams by 10 news services) */
/* */
/* Note: (a) news services rank varying numbers of teams; */
/* (b) not all teams are ranked by all news services; */
/* (c) each team is ranked by at least one service; */
/* (d) rank 20 is missing for UPI. */
/* */
/*-----------------------------------------------------------*/
data HoopsRanks;
input School $13. CSN DurSun DurHer WashPost USAToday
Sport InSports UPI AP SI;
label CSN = 'Community Sports News (Chapel Hill, NC)'
DurSun = 'Durham Sun'
DurHer = 'Durham Morning Herald'
WashPost = 'Washington Post'
USAToday = 'USA Today'
Sport = 'Sport Magazine'
InSports = 'Inside Sports'
UPI = 'United Press International'
AP = 'Associated Press'
SI = 'Sports Illustrated'
;
format CSN--SI 5.1;
datalines;
Louisville 1 8 1 9 8 9 6 10 9 9
Georgia Tech 2 2 4 3 1 1 1 2 1 1
Kansas 3 4 5 1 5 11 8 4 5 7
Michigan 4 5 9 4 2 5 3 1 3 2
Duke 5 6 7 5 4 10 4 5 6 5
UNC 6 1 2 2 3 4 2 3 2 3
Syracuse 7 10 6 11 6 6 5 6 4 10
Notre Dame 8 14 15 13 11 20 18 13 12 .
Kentucky 9 15 16 14 14 19 11 12 11 13
LSU 10 9 13 . 13 15 16 9 14 8
DePaul 11 . 21 15 20 . 19 . . 19
Georgetown 12 7 8 6 9 2 9 8 8 4
Navy 13 20 23 10 18 13 15 . 20 .
Illinois 14 3 3 7 7 3 10 7 7 6
Iowa 15 16 . . 23 . . 14 . 20
Arkansas 16 . . . 25 . . . . 16
Memphis State 17 . 11 . 16 8 20 . 15 12
Washington 18 . . . . . . 17 . .
UAB 19 13 10 . 12 17 . 16 16 15
UNLV 20 18 18 19 22 . 14 18 18 .
NC State 21 17 14 16 15 . 12 15 17 18
Maryland 22 . . . 19 . . . 19 14
Pittsburgh 23 . . . . . . . . .
Oklahoma 24 19 17 17 17 12 17 . 13 17
Indiana 25 12 20 18 21 . . . . .
Virginia 26 . 22 . . 18 . . . .
Old Dominion 27 . . . . . . . . .
Auburn 28 11 12 8 10 7 7 11 10 11
St. Johns 29 . . . . 14 . . . .
UCLA 30 . . . . . . 19 . .
St. Joseph's . . 19 . . . . . . .
Tennessee . . 24 . . 16 . . . .
Montana . . . 20 . . . . . .
Houston . . . . 24 . . . . .
Virginia Tech . . . . . . 13 . . .
;
/* PROC MEANS is used to output a data set containing the */
/* maximum value of each of the newspaper and magazine */
/* rankings. The output data set, maxrank, is then used */
/* to set the missing values to the next highest rank plus */
/* thirty-six, divided by two (that is, the mean of the */
/* missing ranks). This ad hoc method of replacing missing */
/* values is based more on intuition than on rigorous */
/* statistical theory. Observations are weighted by the */
/* number of nonmissing values. */
/* */
title 'Pre-Season 1985 College Basketball Rankings';
proc means data=HoopsRanks;
output out=MaxRank
max=CSNMax DurSunMax DurHerMax
WashPostMax USATodayMax SportMax
InSportsMax UPIMax APMax SIMax;
run;
Output 69.2.1
Summary Statistics for Basketball Rankings Using PROC MEANS
| CSN |
| DurSun |
| DurHer |
| WashPost |
| USAToday |
| Sport |
| InSports |
| UPI |
| AP |
| SI |
|
| Community Sports News (Chapel Hill, NC) |
| Durham Sun |
| Durham Morning Herald |
| Washington Post |
| USA Today |
| Sport Magazine |
| Inside Sports |
| United Press International |
| Associated Press |
| Sports Illustrated |
|
| 30 |
| 20 |
| 24 |
| 19 |
| 25 |
| 20 |
| 20 |
| 19 |
| 20 |
| 20 |
|
| 15.5000000 |
| 10.5000000 |
| 12.5000000 |
| 10.4210526 |
| 13.0000000 |
| 10.5000000 |
| 10.5000000 |
| 10.0000000 |
| 10.5000000 |
| 10.5000000 |
|
| 8.8034084 |
| 5.9160798 |
| 7.0710678 |
| 6.0673607 |
| 7.3598007 |
| 5.9160798 |
| 5.9160798 |
| 5.6273143 |
| 5.9160798 |
| 5.9160798 |
|
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
| 1.0000000 |
|
| 30.0000000 |
| 20.0000000 |
| 24.0000000 |
| 20.0000000 |
| 25.0000000 |
| 20.0000000 |
| 20.0000000 |
| 19.0000000 |
| 20.0000000 |
| 20.0000000 |
|
